Spin up problem and accelerating convergence to steady state

In this study, we investigate the problem of accelerating convergence to equilibrium for the Navier–Stokes equation (NSE) models. The first model is obtained adding the time relaxation term “+κ(u-u¯)”, where u¯ denotes the time filter of u  , to NSE. The second one is the case of adding term “-λΔ(u-u¯)”. Both of the models have the same steady state solution with the NSE. For both models, it is shown that the approximate solutions converge to steady state solution u∞u∞ faster than the solution of the NSE, which means that the solutions of the models reach equilibrium faster. If the approximate solution of the NSE reach to steady state at the time tn+1tn+1, the acceleration to steady state of the approximate solutions of the models can be obtained more faster than the acceleration to steady state of the approximate solution of the NSE for t